A weather-balloon envelope is manufactured in the shape of a right circular cylinder…
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A weather-balloon envelope is manufactured in the shape of a right circular cylinder whose radius r (in cm) and height h (in cm) both vary with time. Helium is pumped into the balloon at a constant rate of 5 liters per second. At the instant when r=7 cm and h=30 cm, the height is increasing at a rate of 0.40 cm/s.
(a) Convert the helium inflow rate to cubic centimeters per second. Show your unit conversion clearly.
(b) Using the cylinder volume formula V=πr2h, determine the instantaneous rate of change of the radius, dr/dt, at the given instant. Present each step of your derivation, keeping units throughout.
(c) The total surface area of the cylinder (including its two circular ends) is A=2πrh+2πr2. Find dA/dt at the same instant and explain the physical significance of its sign.